Subgroups — C₁₃×#13006 #15699345

Order 3328.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#473	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2916	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2117	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#137	—	2	1	no	—
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#568	—	2	1	no	—
11	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₁₃×#473	—	2	2	no	—
15	C₁₃×#473	C₄	1	4	yes	—
16	C₁₃×#430	—	2	1	no	—
17	C₁₃×#430	—	2	2	no	—
18	C₂×C₁₃×#108	—	2	2	no	—
19	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#101	—	2	1	no	—
23	C₁₃×#362	—	2	2	no	—
24	C₁₃×#362	—	2	2	no	—
25	C₁₃×#350	—	2	2	no	—
26	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₁₃	—	2	1	no	—
28	C₁₃×#137	—	2	4	no	—
29	C₁₃×#137	—	4	2	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	4	1	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×SD₁₆	—	4	1	no	—
38	C₁₃×#131	—	2	1	no	—
39	C₁₃×#131	—	2	1	no	—
40	C₁₃×#131	—	4	2	no	—
41	C₁₃×#131	—	4	2	no	—
42	C₁₃×#131	C₂×C₄	1	2	yes	—
43	C₁₃×#131	D₄	1	2	yes	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	—	2	4	no	—
46	C₂×C₁₃×M₁₆	—	2	1	no	—
47	C₂×C₁₃×M₁₆	—	2	1	no	—
48	C₂×C₁₃×M₁₆	—	2	1	no	—
49	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₃	—	2	1	no	—
51	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×M₃₂	—	2	1	no	—
53	C₁₃×M₃₂	—	2	1	no	—
54	C₁₃×M₃₂	—	2	1	no	—
55	C₁₃×M₃₂	—	2	1	no	—
56	C₂×C₁₃×C₁₆	—	2	1	no	—
57	C₂×C₁₃×C₁₆	—	2	1	no	—
58	C₁₃×#108	—	2	4	no	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	—	2	1	no	—
61	C₁₃×#108	—	2	1	no	—
62	C₁₃×#108	C₂×C₄	1	2	yes	—
63	C₁₃×#108	D₄	1	2	yes	—
64	C₁₃×#108	D₄	1	2	yes	—
65	C₁₃×#101	—	2	4	no	—
66	#13006	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₃×#41	—	2	1	no	—
68	C₁₃×#41	—	2	1	no	—
69	C₁₃×#41	C₂²⋊C₄	1	2	yes	—
70	C₁₃×#41	D₈	1	2	yes	—
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂×Q₈×C₁₃	SD₁₆	1	1	yes	characteristic
78	C₂×D₄×C₁₃	—	2	1	no	—
79	C₂×D₄×C₁₃	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	—	2	1	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	4	4	no	—
86	C₁₃×Q₁₆	—	4	1	no	—
87	C₁₃×Q₁₆	—	4	1	no	—
88	C₁₃×SD₁₆	—	4	4	no	—
89	C₁₃×M₁₆	—	2	1	no	—
90	C₁₃×M₁₆	—	2	2	no	—
91	C₁₃×M₁₆	—	2	2	no	—
92	C₁₃×M₁₆	—	2	2	no	—
93	C₁₃×M₁₆	—	2	2	no	—
94	C₁₃×M₁₆	—	2	2	no	—
95	C₁₃×M₁₆	—	4	4	no	—
96	C₁₃×M₁₆	—	4	1	no	—
97	C₁₃×M₁₆	—	4	1	no	—
98	C₁₃×M₁₆	—	4	1	no	—
99	C₁₃×M₁₆	C₂²⋊C₄	1	2	yes	—
100	C₂×C₈×C₁₃	—	2	1	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	4	4	no	—
104	C₂×C₈×C₁₃	—	4	1	no	—
105	C₂×C₈×C₁₃	—	4	1	no	—
106	C₂×C₈×C₁₃	—	4	1	no	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
113	C₁₃×C₁₆	—	4	1	no	—
114	C₁₃×C₁₆	—	4	1	no	—
115	C₂×#473	C₂×C₁₃	1	1	yes	characteristic
116	#2916	C₂×C₁₃	1	1	yes	characteristic
117	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2117	C₂×C₁₃	1	1	yes	characteristic
120	#1388	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#1385	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂³×C₁₃	—	2	1	no	—
123	C₂³×C₁₃	#137	1	1	yes	characteristic
124	Q₈×C₁₃	—	2	1	no	—
125	Q₈×C₁₃	—	4	2	no	—
126	Q₈×C₁₃	—	4	2	no	—
127	Q₈×C₁₃	—	8	1	no	—
128	Q₈×C₁₃	—	8	1	no	—
129	Q₈×C₁₃	#102	1	2	yes	—
130	D₄×C₁₃	—	2	2	no	—
131	D₄×C₁₃	—	2	2	no	—
132	D₄×C₁₃	—	2	1	no	—
133	D₄×C₁₃	#102	1	2	yes	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	8	1	no	—
139	C₂×C₄×C₁₃	—	8	1	no	—
140	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	4	2	no	—
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	—	4	2	no	—
150	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₈×C₁₃	#121	1	1	yes	characteristic
153	C₈×C₁₃	#121	1	1	yes	characteristic
154	C₂×#137	—	2	1	no	—
155	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
156	#568	—	2	1	no	—
157	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
160	#473	—	2	2	no	—
161	#473	C₄×C₁₃	1	4	yes	—
162	#430	—	2	1	no	—
163	#430	—	2	2	no	—
164	C₂×#108	—	2	2	no	—
165	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×#101	—	2	1	no	—
169	#362	—	2	2	no	—
170	#362	—	2	2	no	—
171	#350	—	2	2	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	4	1	no	—
174	C₂²×C₁₃	—	8	1	no	—
175	C₂²×C₁₃	#417	1	1	yes	characteristic
176	C₂²×C₁₃	#448	1	1	yes	characteristic
177	C₂²×C₁₃	#480	1	1	yes	characteristic
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	#386	1	1	yes	characteristic
181	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
182	C₄×C₁₃	#447	1	1	yes	characteristic
183	C₄×C₁₃	#481	1	1	yes	characteristic
184	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
185	C₂²×Q₈	—	2	1	no	—
186	#137	—	2	4	no	—
187	#137	—	4	2	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	—	4	1	no	—
194	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
195	C₂×SD₁₆	—	4	1	no	—
196	#131	—	2	1	no	—
197	#131	—	2	1	no	—
198	#131	—	4	2	no	—
199	#131	—	4	2	no	—
200	#131	C₂×C₄×C₁₃	1	2	yes	—
201	#131	D₄×C₁₃	1	2	yes	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
210	M₃₂	—	2	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	2	1	no	—
213	M₃₂	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	#108	—	2	4	no	—
217	#108	—	2	1	no	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	C₂×C₄×C₁₃	1	2	yes	—
221	#108	D₄×C₁₃	1	2	yes	—
222	#108	D₄×C₁₃	1	2	yes	—
223	#101	—	2	4	no	—
224	C₂×C₁₃	—	2	1	no	—
225	C₂×C₁₃	—	2	1	no	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	—	4	2	no	—
228	C₂×C₁₃	#1874	1	1	yes	characteristic
229	#41	—	2	1	no	—
230	#41	—	2	1	no	—
231	#41	C₁₃×C₂²⋊C₄	1	2	yes	—
232	#41	C₁₃×D₈	1	2	yes	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	C₁₃×SD₁₆	1	1	yes	characteristic
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₁₃×SD₁₆	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	2	2	no	—
247	Q₁₆	—	4	4	no	—
248	Q₁₆	—	4	1	no	—
249	Q₁₆	—	4	1	no	—
250	SD₁₆	—	4	4	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	4	4	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
274	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
275	C₁₆	—	4	1	no	—
276	C₁₆	—	4	1	no	—
277	C₁₃	#13006	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	—	2	1	no	—
279	C₂³	C₁₃×#137	1	1	yes	characteristic
280	Q₈	—	2	1	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	8	1	no	—
284	Q₈	—	8	1	no	—
285	Q₈	C₁₃×#102	1	2	yes	—
286	D₄	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	1	no	—
289	D₄	C₁₃×#102	1	2	yes	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
298	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
299	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
301	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	4	2	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	2	no	—
306	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
307	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
308	C₈	C₁₃×#121	1	1	yes	characteristic
309	C₈	C₁₃×#121	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	C₁₃×#417	1	1	yes	characteristic
314	C₂²	C₁₃×#448	1	1	yes	characteristic
315	C₂²	C₁₃×#480	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	8	2	no	—
318	C₄	C₁₃×#386	1	1	yes	characteristic
319	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
320	C₄	C₁₃×#447	1	1	yes	characteristic
321	C₄	C₁₃×#481	1	1	yes	characteristic
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	4	2	no	—
326	C₂	C₁₃×#1874	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1874
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₁₃×#362	C₂	semidirect
4. C₂×C₁₃×#362	C₂	semidirect
6. C₁₃×#1388	C₂	semidirect
7. C₁₃×#1385	C₂	semidirect
66. #13006	C₁₃	direct (N × H)
117. C₂×#362	C₂×C₁₃	semidirect
118. C₂×#362	C₂×C₁₃	semidirect
120. #1388	C₂×C₁₃	semidirect
121. #1385	C₂×C₁₃	semidirect
277. C₁₃	#13006	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#473	2
C₁₃×#2916	2
C₂×C₁₃×#362	2
C₁₃×#2117	2
C₁₃×#1388	2
C₁₃×#1385	2
#13006	13

Subgroups — C13×#13006 #15699345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13006 #15699345